Omitted variable bias in regression only containing dummy variables What is the right way to think of omitted variable bias in a regression that only has dummy variables?
Let's say I have the following equation:
(1) y=β0+β1x1+β2x2+β3x3+ϵ, 
where y is the price of shoes; and x1, x2 and x3 are dummy variables for three different regions (x4, the reference region was omitted).
And I suspect that I am missing a variable (x4) to adjust for 'level of urbanization of each region' -- each region contains an uneven number of areas with different levels of urbanization. Thus, the true model is, or so I suspect:
(2) y=β0+β1x1+β2x2+β3x3+δx4+ϵ
Now I know I can sign the bias of any one of the coefficients in equation (1) if two conditions are met: a) x4 is correlated with either x1, x2 or x3; and b) x4 has an impact on y (i.e. δ>0).
However, I am not sure if I can explore condition a) above since the correlation of x4, in this case, would be with a categorical variable, not a continuous one.
How can I go about this?
 A: Whether a variable is binary or continuous has no bearing on its potential for omitted variable bias (unless it's a poorly measured version of a truly continuous variable). If the "true" model uses the binary variables as they are, then the same rules that apply for continuous predictors and continuous omitted variables apply here. Regression models don't know whether predictors are continuous or binary; they treats them all as continuous. It is only our interpretation of them as binary that makes them any different from continuous variables.
Again, things get hairy if the binary variable is a categorized version of a continuous variable (or has measurement error in it some other way). 
A: You're right that the requirement is $\mathrm{cov}\left(x_4,x_1\right)\neq0$. The important part is that $\mathrm{cov}$ doesn't care if any of the variables (or both) is continuous or categorical. You can calculate it, irrespectively of what is their nature!
Concretely in your case: if $x_1$ is binary, and $x_4$ is continuous then
$\mathrm{cov}\left(x_4,x_1\right)=\frac{1}{n}\sum_{i=1}^n \left(x_{1i}-\overline{x_1}\right)\left(x_{4i}-\overline{x_4}\right),$
i.e. the definition is totally the same. What you have to note in this formula, is that you can calculate everything here, it doesn't matter that $x_1$ is binary!
Yes, the formula will become
$\mathrm{cov}\left(x_4,x_1\right)=\frac{1}{n}\sum_{i=1}^n \left(x_{1i}-\overline{x_1}\right)\left(x_{4i}-\overline{x_4}\right)=\frac{-\overline{x_1}\sum_{x_{1i}=0}\left(x_{4i}-\overline{x_4}\right) + \left(1-\overline{x_1}\right)\sum_{x_{1i}=1}\left(x_{4i}-\overline{x_4}\right)}{n},$
but that is just a technical question (simplification).
A: If we focus on the primary purpose of a regression model as summarizing mean differences, it's easy to see there's no such thing as an omitted variable bias. In fact, some causal schools of thought (such as the one to which I belong) think that so long as there is any residual standard error, then variables have been omitted (no such thing as inherent randomness). All "omitted variable bias" amounts to is an imprecise catch-all which lumps together confounding and subpar predictions. 
If your conditional mean response is summarized by 3 fixed effects for the four-level region variable, what you have is a simple kind of ANOVA. If you add the between-region adjustment variable of urbanization, you have a special type of case-mix-adjusted model. Some theory shows that the added adjustment changes the interpretation of the fixed effects for region by standardizing for urbanization. What you get is a summary of mean differences between region that is independent of urbanization. How you interpret that depends on how urbanization is calculated. This enables thought-experiment comparisons such as what would be the expected response in region A if it had the level of urbanization of region B? (be careful because a lot of thought experiments are useless if they are not part of a prespecified analysis plan).
So no you haven't omitted any variable, and your result is not biased, you just have a model that summarizes all available information. I would reserve a discussion about "urbanization" to contextualize the different ways it could be measured, what is known about the regions, and implications for further research.
